arXiv research

In this paper we prove that every proper Lie groupoid admits a desingularization to a regular proper Lie groupoid. When equipped with a Riemannian metric, we show that it admits a desingularization to a regular Riemannian proper Lie groupoid, arbitrarily close to the original one in the Gromov-Hausdorff distance betwee…

New obstructions found for smooth desingularization of compact Einstein orbifolds.

Study on Einstein deformations and desingularizations, finding some 4-orbifolds cannot be limits of smooth metrics.

We introduce and study a "desingularization" of a Lie groupoid $G$ along an "$A(G)$-tame" submanifold $L$ of the space of units $M$. An $A(G)$-tame submanifold $L \subset M$ is one that has, by definition, a tubular neighborhood on which $A(G)$ becomes a thick pull-back Lie algebroid. The construction of the desingular…

Desingularizes singular spaces using sheaves and groupoids.

We prove that desingularizations of non degenerate Poincaré-Einstein metrics with A1 singularities remain non degenerate. In principle this enables a recursive procedure to desingularize the other Fuchsian singularities. We illustrate this procedure by the A2 case.

New proof of Grauert's theorem using differential geometry.

We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth sp…

Complete minimal surfaces with any genus found in a specific 3-manifold.

In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map. We also compute some of these products and prove the nontriviality of the desingularization morphism.

We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for defor…

This paper is concerned with the existence of constant scalar curvature Kaehler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kaehler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already car…

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are …

Smoothly attaches manifolds with controlled curvature.

In the first part of the paper we discuss the current status of the application of the gluing methodology to doubling and desingularization constructions for minimal surfaces in Riemannian three-manifolds. In particular a doubling construction for equatorial spheres in $S^3(1)$ is announced. Aspects of the current unde…

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted …

Study X-ray transform on manifolds, desingularize, and improve mapping properties.

This paper is a follow-up to an earlier paper math.DG/0410260 on desingularizations of Calabi-Yau 3-folds with a conical singularity. In math.DG/0410260 we study Calabi-Yau 3-folds M_0 with a conical singularity at x modelled on some Calabi-Yau cone V, and construct a desingularization of M_0 by gluing in an Asymptotic…

Constructing translating solitons from Lagrangian Grim Reapers.

This study addresses transitions in conically singular associative submanifolds and their desingularizations.

For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of π/m. They are invariant under a finite group of screw motions and have no reflection symmetry across a great…

New minimal surfaces desingularize three Clifford tori, proving uniqueness and characterizing them.

This article is devoted to the study of smooth desingularization, which are customary employed in the definition of De Rham Intersection Cohomology with differential forms. In this paper we work with the category of Thom-Mather simple spaces. We construct a functor which sends each Thom-Mather simple space into a smoot…

For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with …

Desingularizes singular foliations with a locally compact groupoid.

Consider a singular Riemannian foliation (s.r.f for short) on a compact manifold. By successive blow-ups along the strata, we construct a regular Riemannian foliation on another compact Riemannian manifold and a desingularization map that projects leaves of the regular Riemannian foliation into leaves of the s.r.f. Thi…

Building upon ideas of Hironaka, Bierstone-Milman, Malgrange and others we generalize the inverse and implicit function theorem (in differential, analytic and algebraic setting) to sets of functions of larger multiplicities (or ideals). This allows one to describe singularities given by a finite set of generators or by…

We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on…

Consider an Einstein orbifold $(M_0,g_0)$ of real dimension $2n$ having a singularity with orbifold group the cyclic group of order $n$ in ${\rm{SU}}(n)$ which is generated by an $n$th root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is…

Desingularizes conically singular Cayley submanifolds.

We desingularize a branch point $p$ of a minimal disk $F_0(\mathbb{D})$ in $\mathbb{R}^4$ through immersions $F_t$'s which have only transverse double points and are branched covers of the plane tangent to $F_0(\mathbb{D})$ at $p$. If $F_0$ is a topological embedding and thus defines a knot in a sphere/cylinder around …

We use Bryant Representation to construct constant mean curvature one surfaces in hyperbolic space that desingularize a horosphere packing.

Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma…

We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the…

By results of the author there exists a projective (holomorphic) symplectic desingularization of the moduli space of rank-two torsion-free sheaves on a genus-two Jacobian with $c_1=0$ and $c_2=2$. This desingularization has a natural map to the self-product of the Jacobian. We show that the fiber over $(0,0)$ is a 6-di…

Given a coassociative 4-fold N with a conical singularity in a varphi-closed 7-manifold M (a manifold endowed with a distinguished closed 3-form varphi), we construct a smooth family, {N'(t): t\in(0,tau)} for some tau>0, of (smooth, nonsingular,) compact coassociative 4-folds in M which converge to N in the sense of cu…

We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat K{ä}hler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc 4-manifolds with metric singularities along points and embedded circles…

In 1996 M. Traizet obtained singly periodic minimal surfaces with Scherk ends of arbitrary genus by desingularizing a set of vertical planes at their intersections. However, in Traizet's work it is not allowed that three or more planes intersect at the same line. In our paper, by a {\it saddle-tower} we call the desing…

We deform a minimal disk in $\mathbb{R}^4$ with a branch point into symplectic minimally immersed disks with only transverse double points.

We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces... In this work, we prove existence o…

A steady state (or equilibrium point) of a dynamical system is hyperbolic if the Jacobian at the steady state has no eigenvalues with zero real parts. In this case, the linearized system does qualitatively capture the dynamics in a small neighborhood of the hyperbolic steady state. However, one is often forced to consi…

Constructs CMC hypersurfaces in S^4 from piecewise-smooth unions of spheres.

The purpose of this paper is to generalize the regular Optimal Reduction Theorem to general proper Dirac actions, formulated both in terms of point and orbit reduction. A comparison to general standard singular Dirac reduction is given emphasizing the desingularization role played by optimal reduction.

For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.

Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow

We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expa…

A smooth map between smooth manifolds is called a special generic map if it has only definite fold points as its singularities. In this paper, we give conditions for a special generic map into the 3-dimensional Euclidean space to be factored as the composition of an embedding and a projection for certain dimensions.